Elements of Algebra

428 ELEMENTS PART II. only be divisible by 3, and not by 9; consequently, also, $p^2 + 3q^2$ could only be divisible by 3, and not by 9, and therefore could not be a square; so that $p$ cannot be divisible by 3, and $p^2$ will be a number of the class $3n + 1$. 5. Since $p$ is not divisible by 3, $q$ must be so; for otherwise $q^2$ would be a number of the class $3n+1$, and consequently $p^2 + q^2$ a number of the class $3n+2$, which cannot be a square: therefore $q$ must be divisible by 3. 6. Nor is $p$ divisible by 5; for if that were the case, $q$ would not be so, and $q^2$ would be a number of the class $5n+1$, or $5n+4$; consequently, $3q^2$ would be of the class $5n + 3$, or $5n+ 2$; and as $p^2 + 3q^2$ would belong to the same classes, this formula therefore could not in that case become a square; consequently $p$ must not be divisible by 5, and $p^2$ must be a number of the class $5n + 1$, or of the class $5n + 4$. 7. But since $p$ is not divisible by 5, let us see whether $q$ is divisible by 5, or not; since if $q$ were not divisible by 5, $q^2$ must be of the class $5n + 2$, or $5n + 3$, as we have already seen; and since $p^2$ is of the class $5n+1$, or $5n + 4$, $p^2+3q^2$ must be the same; namely, $5n+1$, or $5n+4$; and therefore, of one of the forms $5n + 3$, or $5n+ 2$. Let us consider these cases separately. If we suppose $p^2 (F) 5n+1*$, then we must have $q^2 (F) 5n+4$, because otherwise $p^2 + q^2$ could not be a square; but we should then have $3q^2 (F) 5n+2$ and $p^2 + 3q^2 (F) 5n+3$, which cannot be a square. In the second place, let $p^2 (F) 5n+4$; in this case we must have $q^2 (F) 5n+1$, in order that $p^2 + q^2$ may be a square, and $3q^2 (F) 5n+3$; therefore $p^2+3q^2 (F) 5n + 2$, which cannot be a square. It follows, therefore, that $q$ must be divisible by 5. 8. Now, $q$ being divisible first by 4, then by 3, and in the third place by 5, it must be such a number as $4 \times 3 \times 5m$, or $q = 60m$; so that our formulæ would become $p^2 + 3600m^2 = r^2$, and $p^2 + 10800m^2 = s^2$: this being established, the first, subtracted from the second, will give $7200m^2 = s^2 - r^2 = (s + r) \times (s - r)$; so that $s + r$ and $s - r$ must be factors of $7200m^2$, and at the same time * In the former editions of this work, the sign = is used to express the words, “of the form.” This was adopted in order to save the repetition of these words; but as it may occasionally produce ambiguity, or confusion, it was thought proper to substitute (F) instead of =, which is to be read thus: $p^2(F)5n+1$, of the form $5n+1$.